Connecting Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe the behavior of functions as inputs approach positive or negative infinity, determining horizontal asymptotes.

Limits10–12% of exam

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Context

What this topic is and why it exists

Limits at infinity describe the behavior of functions as x approaches positive or negative infinity.

You're looking at how outputs stabilize, not what they exactly become.

This is where horizontal asymptotes come in.

If a function approaches a constant value as x heads toward infinity, that value is a horizontal asymptote.

The trap here is thinking that every function has one.

Some functions, like polynomials with higher degree terms, don't settle into a horizontal line.

You think they're going to flatten out, but they just keep growing.

Rational functions are where you'll see horizontal asymptotes most often.

If the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.

If the numerator's degree is less, the asymptote is y = 0.

Get this wrong, and you're misreading the long-term behavior.

Limits at infinity aren't just about finding asymptotes; they're about understanding how functions behave when x is enormous.

That understanding is the real point.

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